Observe that by the boundedness of \({\mathcal {F}}\)  we have \(\mathbf {a}\in \mathrm {int}\left({\mathcal {K}}^*\right)\)  which implies that \(\mathbf {v}\mathbf {a}= (\mathbf {v})>0\ \forall \mathbf {v}\in {\mathcal {K}}\)  so that \(\mathbf {a}\mathbf {a}\mathrm {int}\mathcal {CPP}({\mathcal {K}})^*\) , hence \(\mathbf {a}\mathbf {a}{\mathsf {V}}>0\ \forall \, {\mathsf {V}}\in \mathcal {CPP}({\mathcal {K}})\) . We argue that therefore the extreme points of \({\mathcal {F}}_{Shor}\)  are precisely the intersections of the extreme rays of \(\mathcal {CPP}({\mathcal {K}})\)  and the hyperplane \({\mathcal {H}}\left\lbrace {\mathsf {X}}\in {\mathcal {S}}^n\colon \mathbf {a}\mathbf {a}{\mathsf {X}}= 1 \right\rbrace \) , so that these extreme points are actually rank-1 matrices. Indeed if \({\mathsf {Y}}\in \mathrm {ext}{\mathcal {F}}_{shor}\)  but \(\left\lbrace \lambda {\mathsf {Y}}\colon \lambda \ge 0 \right\rbrace \)  was not an extreme ray of \(\mathcal {CPP}({\mathcal {K}})\)  then by [1]} \({\mathsf {Y}}= \sum _{i=1}^{k}\lambda _i{\mathsf {Y}}_i,\ \sum _{i=1}^{k}\lambda _i = 1,\ \lambda _i\ge 0, \ i\in [1\!:\!k]\)  for some generators of extreme rays \({\mathsf {Y}}_i\in \mathrm {ext}\mathcal {CPP}({\mathcal {K}}),\ i \in [1\!:\!k]\) . All these extreme rays have to meet the hyperplane \({\mathcal {H}}\)  since \(\mathbf {a}\mathbf {a}\mathrm {int}\mathcal {COP}({\mathcal {K}})\)  so that there are \(\mu _i>0\)  with \(\mathbf {a}\mathbf {a}\left(\mu _i{\mathsf {Y}}_i\right) = 1\)  so acutally \(\mu _i = 1/\left(\mathbf {a}\mathbf {a}{\mathsf {Y}}_i\right)\) . But this implies \({\mathsf {Y}}= \sum _{i=1}^{k}\lambda _i{\mathsf {Y}}_i = \sum _{i=1}^{k}\tfrac{\lambda _i}{\mu _i}\mu _i{\mathsf {Y}}_i\)  where \(\mu _i{\mathsf {Y}}_i \in {\mathcal {F}}_{Shor}\)  and \(\sum _{i=1}^{k}\tfrac{\lambda _i}{\mu _i} = \sum _{i=1}^{k}\lambda _i\left(\mathbf {a}\mathbf {a}{\mathsf {Y}}_i\right) = \mathbf {a}\mathbf {a}\sum _{i=1}^{k}\lambda _i{\mathsf {Y}}_i =\mathbf {a}\mathbf {a}{\mathsf {Y}}= 1\)  so that \({\mathsf {Y}}\)  cannot be an extreme point of \({\mathcal {F}}_{Shor}\) . Conversely, all extreme rays of \(\mathcal {CPP}({\mathcal {K}})\)  eventually meet \({\mathcal {H}}\)  at a single point which is of the form \(\mathbf {y}\mathbf {y} for some \) yK\( which are extreme in \) FShor\( since rank-1 matrices cannot be obtained as a convex combination of other rank-1 matrices. \)
